To find the image of the circle under the transformation $w = z + 2 + 4i$, we follow these steps:

1. Equation of the Circle:

The given equation is $|z| = 1$, which represents a circle with radius 1 centered at the origin in the complex plane.

2. Transformation:

The transformation $w = z + 2 + 4i$ can be interpreted as a translation of the complex number $z$ by the vector $2 + 4i$. This will move every point on the circle by that amount.

3. Apply the Transformation:

Since $w = z + 2 + 4i$, the transformation simply shifts the center of the circle from the origin to the point $2 + 4i$. Therefore, the radius remains the same, but the center is now at $(2, 4)$.

4. Result:

The image of the circle $|z| = 1$ under the transformation $w = z + 2 + 4i$ is another circle with:

• Center: $(2, 4)$
• Radius: $1$

Final Equation of the Circle:

The equation of the new circle in the complex plane is $|w - (2 + 4i)| = 1$, which represents the image of the original circle after the transformation.

Would you like further details on any steps?

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Here are five related questions to explore further:

1. What is the geometric meaning of a Möbius transformation?
2. How do you find the image of a line under complex transformations?
3. What happens to circles under general linear transformations?
4. Can we generalize this transformation to $w = az + b$?
5. How do transformations affect the curvature of objects?

Tip: Translations in the complex plane are simple shifts of all points by a fixed vector. They do not change the size or shape of geometric objects like circles.